Students use patterns, relationships and functions to model, represent and analyze problems that involve variables. Students also analyze, model and solve problems using various representations i.e., tables, graphs and equations.
By the end of the 3-4 program:
Analyze and extend patterns, and describe the rule in words.
Grade 4
Represent and analyze patterns and functions using words, tables and graphs.
Use patterns to make predictions, identify relationships, and solve problems.
Grade 4
Use models and words to describe, extend and make generalizations of patterns and relationships occurring in computation, numerical patterns, geometry, graphs and other applications.
Write and solve open sentences and explain strategies.
Grade 4
Represent mathematical relationships with equations or inequalities.
Represent an unknown quantity as a variable using a symbol, including letters.
Grade 4
Represent and analyze patterns and functions using words, tables and graphs.
Use variables to create and solve equations representing problem situations.
Grade 4
Use rules and variables to describe patterns and other relationships.
Construct and use a table of values to solve problems associated with mathematical relationships.
Grade 4
Construct a table of values to solve problems associated with a mathematical relationship.
Describe how a change in one variable affects the value of a related variable.
Grade 4
Describe how a change in one variable affects the value of a related variable.
as one increases the other increases or as one increases the other decreases
By the end of the 5-7 program:
Describe, extend and determine the rule for patterns and relationships occurring in numeric patterns, computation, geometry, graphs and other applications.
Grade 5
Justify a general rule for a pattern or a function by using physical materials, visual representations, words, tables or graphs.
Use calculators or computers to develop patterns, and generalize them using tables and graphs.
Grade 6
Represent and analyze patterns, rules and functions, using physical materials, tables and graphs.
Use words and symbols to describe numerical and geometric patterns, rules and functions.
Represent, analyze and generalize a variety of patterns and functions with tables, graphs, words and symbolic rules.
Grade 5
Use variables as unknown quantities in general rules when describing patterns and other relationships.
Grade 7
Represent and analyze patterns, rules and functions with words, tables, graphs and simple variable expressions.
Generalize patterns by describing in words how to find the next term.
Use variables to create and solve equations and inequalities representing problem situations.
Grade 5
Create and interpret the meaning of equations and inequalities representing problem situations.
Grade 6
Produce and interpret graphs that represent the relationship between two variables.
Evaluate simple expressions by replacing variables with given values, and use formulas in problem-solving situations.
Use symbolic algebra to represent and explain mathematical relationships.
Grade 6
Recognize and generate equivalent forms of algebraic expressions, and explain how the commutative, associative and distributive properties can be used to generate equivalent forms.
perimeter as 2(1 + w) or 21 + 2w.
Grade 7
Recognize a variety of uses for variables.
placeholder for an unknown quantity in an equation, generalization for a pattern, formula
Use rules and variables to describe patterns, functions and other relationships.
Grade 5
Use variables as unknown quantities in general rules when describing patterns and other relationships.
Grade 6
Use words and symbols to describe numerical and geometric patterns, rules and functions.
Grade 7
Recognize and explain when numerical patterns are linear or nonlinear progressions.
1,3,5,7... is linear
1,3,4,8,16... is nonlinear
Use representations, such as tables, graphs and equations, to model situations and to solve problems, especially those that involve linear relationships.
Grade 5
Model problems with physical materials and visual representations, and use models, graphs and tables to draw conclusions and make predictions.
Grade 7
Represent linear equations by plotting points in the coordinate plane.
Represent inequalities on a number line or a coordinate plane.
Write, simplify and evaluate algebraic expressions.
Grade 5
Use variables as unknown quantities in general rules when describing patterns and other relationships.
Grade 6
Evaluate simple expressions by replacing variables with given values, and use formulas in problem-solving situations.
Grade 7
Represent and analyze patterns, rules and functions with words, tables, graphs and simple variable expressions.
Justify that two forms of an algebraic expression are equivalent, and recognize when an expression is simplified.
4m = m + m + m + m
a · 5 + 4 = 5a + 4
Solve linear equations and inequalities symbolically, graphically and numerically.
Grade 6
Solve simple linear equations and inequalities using physical models, paper and pencil, tables and graphs.
Grade 7
Create visual representations of equation-solving processes that model the use of inverse operations.
Explain how inverse operations are used to solve linear equations.
Create visual representations of equation-solving processes that model the use of inverse operations.
Use formulas in problem-solving situations.
Grade 5
Use strategies to develop formulas for determining perimeter and area of triangles, rectangles and parallelograms, and volume of rectangular prisms. Measurement
Grade 6
Use strategies to develop formulas for finding circumference and area of circles, and to determine the area of sectors. Measurement
1/2 circle, 2/3 circle, 1/3 circle, 1/4 circle
Evaluate simple expressions by replacing variables with given values, and use formulas in problem-solving situations.
Grade 7
Use formulas in problem-solving situations.
Use strategies to develop formulas for finding area of trapezoids and volume of cylinders and prisms. Measurement
Use and demonstrate understanding of the properties of triangles. Geometry
Use Pythagorean Theorem to solve problems involving right triangles
Use triangle angle sum relationships to solve problems
Graph linear equations and inequalities.
Grade 5
Model problems with physical materials and visual representations, and use models, graphs and tables to draw conclusions and make predictions.
Grade 6
Solve simple linear equations and inequalities using physical models, paper and pencil, tables and graphs.
Produce and interpret graphs that represent the relationship between two variables.
Grade 7
Represent linear equations by plotting points in the coordinate plane.
Represent inequalities on a number line or a coordinate plane.
Analyze functional relationships, and explain how a change in one quantity results in a change in the other.
Grade 5
Describe how the quantitative change in a variable affects the value of a related variable.
describe how the rate of growth varies over time, based upon data in a table or graph
Grade 6
Identify and describe situations with constant or varying rates of change, and compare them.
Grade 7
Analyze linear and simple nonlinear relationships to explain how a change in one variable results in the change of another.
Approximate and interpret rates of change from graphical and numerical data.
Grade 6
Use technology to analyze change.
use computer applications or graphing calculators to display and interpret rate of change
Grade 7
Use graphing calculators or computers to analyze change.
distance-time relationships
By the end of the 8-10 program:
Generalize and explain patterns and sequences in order to find the next term and the nth term.
Grade 8
Generalize patterns and sequences by describing how to find the nth term.
Grade 9
Generalize patterns using functions or relationships (linear, quadratic and exponential), and freely translate among tabular, graphical and symbolic representations.
Identify and classify functions as linear or nonlinear, and contrast their properties using tables, graphs or equations.
Grade 8
Identify functions as linear or nonlinear based on information given in a table, graph or equation.
Grade 9
Define function with ordered pairs in which each domain element is assigned exactly one range element.
Describe problem situations (linear, quadratic and exponential) by using tabular, graphical and symbolic representations.
Translate information from one representation (words, table, graph or equation) to another representation of a relation or function.
Grade 8
Relate the various representations of a relationship.
relate a table to graph, description and symbolic form
Grade 9
Generalize patterns using functions or relationships (linear, quadratic and exponential), and freely translate among tabular, graphical and symbolic representations.
Use algebraic representations, such as tables, graphs, expressions, functions and inequalities, to model and solve problem situations.
Grade 8
Extend the uses of variables to include co-variants where y depends on x.
Use physical models to add and subtract monomials and polynomials, and to multiply a polynomial by a monomial.
Use symbolic algebra (equations and inequalities), graphs and tables to represent situations and solve problems.
Write, simplify and evaluate algebraic expressions (including formulas) to generalize situations and solve problems.
Grade 9
Use formulas to solve problems involving exponential growth and decay.
Add, subtract, multiply and divide monomials and polynomials (division of polynomials by monomials only).
Simplify rational expressions by eliminating common factors and applying properties of integer exponents.
Analyze and compare functions and their graphs using attributes, such as rates of change, intercepts and zeros.
Grade 8
Describe the relationship between the graph of a line and its equation, including being able to explain the meaning of slope as a constant rate of change and y-intercept in real-world problems.
Grade 9
Demonstrate the relationship among zeros of a function, roots of equations, and solutions of equations graphically and in words.
Describe and compare characteristics of the following families of functions: linear, quadratic and exponential functions.
general shape
number of roots
domain
range
rate of change
maximum or minimum
Solve and graph linear equations and inequalities.
Grade 8
Use symbolic algebra (equations and inequalities), graphs and tables to represent situations and solve problems.
Solve linear equations and inequalities graphically, symbolically and using technology.
Grade 9
Write and use equivalent forms of equations and inequalities in problem situations.
changing a linear equation to the slope-intercept form
Find linear equations that represent lines that pass through a given set of ordered pairs, and find linear equations that represent lines parallel or perpendicular to a given line through a specific point.
Solve quadratic equations with real roots by graphing, formula and factoring.
Grade 8
Solve simple quadratic equations graphically.
y = x2 – 16
Grade 9
Solve quadratic equations with real roots by factoring, graphing, using the quadratic formula and with technology.
Solve systems of linear equations involving two variables graphically and symbolically.
Grade 8
Solve 2 by 2 systems of linear equations graphically and by simple substitution.
Interpret the meaning of the solution of a 2 by 2 system of equations.
point, line, no solution
Grade 9
Solve and interpret the meaning of 2 by 2 systems of linear equations graphically, by substitution and by elimination, with and without technology.
Model and solve problem situations involving direct and inverse variation.
Grade 8
Differentiate and explain types of changes in mathematical relationships
linear vs. nonlinear, continuous vs. noncontinuous, direct variation vs. inverse variation
Grade 9
Model and solve problems involving direct and inverse variation using proportional reasoning.
Describe the relationship between slope and the graph of a direct variation and inverse variation.
Describe and interpret rates of change from graphical and numerical data.
Grade 8
Compute and interpret slope, midpoint and distance given a set of ordered pairs.
Describe and compare how changes in an equation affects the related graphs.
for a linear equation changing the coefficient of x affects the slope and changing the constant affects the intercepts
Use graphing calculators or computers to analyze change.
interest compounded over time as a nonlinear growth pattern
Grade 9
Describe how a change in the value of a constant in a linear or quadratic equation affects the related graphs.
Patterns, Functions and Algebra Standard
Students use patterns, relationships and functions to model, represent and analyze problems that involve variables. Students also analyze, model and solve problems using various representations i.e., tables, graphs and equations.By the end of the 3-4 program:
By the end of the 5-7 program:
By the end of the 8-10 program:
Grade 6:
Number Cruncher Activity
Grade 9:
Multiple Linear Regression Activity